Generalized Lagrange functions and weighting coefficient formulae for the harmonic differential quadrature method

2003 ◽  
Vol 57 (3) ◽  
pp. 415-440 ◽  
Author(s):  
T. C. Fung
Keyword(s):  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Weighting Coefficient ◽  
Quadrature Method ◽  
Harmonic Differential Quadrature Method

An approach in deformation and stress analysis of elasto-plastic sandwich cylindrical shell panels based on harmonic differential quadrature method

2016 ◽  
Vol 19 (2) ◽  
pp. 167-191 ◽  
Author(s):  
H Shokrollahi ◽  
F Fallah ◽  
MH Kargarnovin
Keyword(s):  
Cylindrical Shell ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Core Material ◽  
Quadrature Method ◽  
Sandwich Shell ◽  
Finite Element Software ◽  
The Core ◽  
Harmonic Differential Quadrature Method ◽  
Shell Panel

Using harmonic differential quadrature method, an approach to analyze sandwich cylindrical shell panels with any sort of boundary conditions under a generally distributed static loading, undergoing elasto-plastic deformation is proposed. The faces of the sandwich shell panel are made of some isotropic materials with linear work hardening behavior while the core is assumed to be an isotropic material experiencing only elastic behavior. The faces are modeled as thin cylindrical shells obeying the Kirchhoff–Love assumptions. For the core material, it is assumed to be thick and the in-plane stresses are negligible. Upon application of an inner and outer general lateral loading, the governing equations are derived using the principle of virtual displacements. Using an iterative approach, named elasto-plastic harmonic differential quadrature method (EP-HDQM), the equations are solved. The obtained results are compared with the results from finite element software Ansys for different sandwich shell panel configurations. Then, the effects of changing different parameters on the stress and displacement components of sandwich cylindrical shell panels in different elasto-plastic conditions are investigated.

An Differential Quadrature Finite Element and the Differential Quadrature Hierarchical Finite Element Methods for the Dynamics Analysis of on Board Shaft

2021 ◽  
Author(s):  
Saimi Ahmed ◽  
Hadjoui Abdelhamid ◽  
Bensaid Ismail ◽  
Fellah Ahmed
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Finite Element Methods ◽  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Weighting Coefficient ◽  
Quadrature Method ◽  
Hierarchical Finite Element Method ◽  
Element Method ◽  
Hierarchical Finite Element

In this paper the dynamic analysis of a shaft rotor whose support is mobile is studied. For the calculation of kinetic energy and stiffness energy, the beam theory of Euler Bernoulli was used, and the matrices of elements and systems are developed using two methods derived from the differential quadrature method (DQM). The first method is the Differential Quadrature Finite Element Method (DQFEM) systematically, as a combination of the Differential Quadrature Method (DQM) and the Standard Finite Element Method (FEM), which has a reduced computational cost for problems in dynamics. The second method is the Differential Quadrature Hierarchical Finite Element Method (DQHFEM) which is used by expressing the matrices of the hierarchical finite element method in a similar form to that of the Differential Quadrature Finite Element Method and introducing an interpolation basis on the element boundary of the hierarchical finite element method. The discretization element used for both methods is a three-dimensional beam element. In the differential quadrature finite element method (DQFEM), the mass, gyroscopic and stiffness matrices are simply calculated using the weighting coefficient matrices given by the differential quadrature (DQ) and Gauss-Lobatto quadrature rules. The sampling points are determined by the Gauss-Lobatto node method. In the Differential Quadrature Hierarchical Finite Element Method (DQHFEM) the same approaches were used, and the cubic Hermite shape functions and the special Legendre polynomial Rodrigues shape polynomial were added. The assembly of the matrices for both methods (DQFEM and DQHFEM) is similar to that of the classical finite element method. The results of the calculation are validated with the h- and hp finite element methods and also with the literature.

Harmonic Differential Quadrature Method for Surface Location Error Prediction and Machining Parameter Optimization in Milling

2015 ◽  
Vol 137 (2) ◽  
Author(s):  
Ye Ding ◽  
XiaoJian Zhang ◽  
Han Ding
Keyword(s):  
Differential Quadrature Method ◽  
Differential Quadrature ◽  
Machining Parameters ◽  
Quadrature Method ◽  
Location Error ◽  
Sampling Grid ◽  
Surface Location Error ◽  
Harmonic Differential Quadrature Method ◽  
Surface Location ◽  
Grid Points

This paper presents a semi-analytical numerical method for surface location error (SLE) prediction in milling processes, governed by a time-periodic delay-differential equation (DDE) in state-space form. The time period is discretized as a set of sampling grid points. By using the harmonic differential quadrature method (DQM), the first-order derivative in the DDE is approximated by the linear sums of the state values at all the sampling grid points. On this basis, the DDE is discretized as a set of algebraic equations. A dynamic map can then be constructed to simultaneously determine the stability and the steady-state SLE of the milling process. To obtain optimal machining parameters, an optimization model based on the milling dynamics is formulated and an interior point penalty function method is employed to solve the problem. Experimentally validated examples are utilized to verify the accuracy and efficiency of the proposed approach.

scholarly journals Buckling Analyses of Axially Functionally Graded Nonuniform Columns with Elastic Restraint Using a Localized Differential Quadrature Method

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Yasin Yilmaz ◽  
Zekeriya Girgin ◽  
Savas Evran
Keyword(s):  
Eigenvalue Problems ◽  
Domain Boundary ◽  
Differential Quadrature Method ◽  
Axial Direction ◽  
Functionally Graded ◽  
Differential Quadrature ◽  
Weighting Coefficient ◽  
Physical Domain ◽  
Quadrature Method ◽  
Central Type

A localized differential quadrature method (LDQM) is introduced for buckling analysis of axially functionally graded nonuniform columns with elastic restraints. Weighting coefficients of differential quadrature discretization are obtained making use of neighboring points in forward and backward type schemes for the reference grids near the beginning and end boundaries of the physical domain, respectively, and central type scheme for the reference grids inside the physical domain. Boundary conditions are directly implemented into weighting coefficient matrices, and there is no need to use fictitious points near the boundaries. Compatibility equations are not required because the governing differential equation is discretized only once for each reference grid using neighboring points and variation of flexural rigidity is taken to be continuous in the axial direction. A large case of columns having different variations of cross-sectional profile and modulus of elasticity in the axial direction are considered. The results for nondimensional critical buckling loads are compared to the analytical and numerical results available in the literature. Some new results are also given. Comparison of the results shows the potential of the LDQM for solving such generalized eigenvalue problems governed by fourth-order variable coefficient differential equations with high accuracy and less computational effort.

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